Introduction to Cosmology

128 Thermal History of the Universe

Also, the muons decay fast compared with the age of the Universe, with a lifetime

of 2.2μs, by the processes

휇−→e−+휈e+휈휇,휇+→e++휈e+휈휇. (6.53)

Almost the entire mass of the muon, or 105.7MeV, is available as kinetic energy to

the final state particles. This is the reason for its short mean life. Here again the con-

servation of lepton numbers, separately for the e-family and the휇-family, is observed.

Below the muon mass (actually at about 50MeV), the temperature in the Universe

cools below the threshold for muon-pair production:

e−+e+→훾virtual→휇++휇−. (6.54)

The time elapsed is less than a millisecond. When the muons have disappeared, we

can reduce푔∗by^72 to^434.

From the reactions in Equations (6.51) and (6.53) we see that the end products

of pion and muon decay are stable electrons and neutrinos. The lightest neutrino휈 1

is certainly stable, and the same is probably also true for휈 2 and휈 3. When this has

taken place we are left with those neutrinos and electrons that only participate in

weak reactions, with photons and with a very small number of nucleons. The number

density of each lepton species is about the same as that of photons.

6.4 Photon and Lepton Decoupling

The considerations about which particles participate in thermal equilibrium at a given

time depend on two timescales: thereaction rateof the particle, taking into account

the reactions which are possible at that energy, and theexpansion rateof the Universe.

If the reaction rate is slow compared with the expansion rate, the distance between

particles grows so fast that they cannot find each other.

Reaction Rates. The expansion rate is given by퐻=푎̇∕푎, and its temperature depen-

dence by Equations (6.44) and (6.45). The average reaction rate can be written

훤=⟨Nv휎(퐸)⟩, (6.55)

where휎(퐸)is the reaction cross-section. The product of휎(퐸)and the velocity푣of
the particle varies over the thermal distribution, so one has to average over it, as is
indicated by the angle brackets. Multiplying this product by the number density푁of
particles per m^3 , one obtains the mean rate훤of reacting particles per second, or the
mean collision time between collisions,훤−^1.
The weak interaction cross-section turns out to be proportional to푇^2 ,

휎≃

퐺^2 F(kT)^2

휋(ℏ푐)^4

, (6.56)

where퐺Fis theFermi couplingmeasuring the strength of the weak interaction. The

number density of the neutrinos is proportional to푇^3 according to Equations (6.12)

and (6.39). The reaction rate of neutrinos of all flavours then falls with decreasing

temperature as푇^5.